PHYSICAL REVIEW B 90, 155316 (2014)
Weyl semimetals from noncentrosymmetric topological insulators
Jianpeng Liu and David Vanderbilt
Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854-8019, USA
(Received 22 September 2014; published 28 October 2014)
We study the problem of phase transitions from three-dimensional topological to normal insulators without
inversion symmetry. In contrast with the conclusions of some previous work, we show that a Weyl semimetal
always exists as an intermediate phase regardless of any constriant from lattice symmetries, although the interval
of the critical region is sensitive to the choice of path in the parameter space and can be very narrow. We
demonstrate this behavior by carrying out first-principles calculations on the noncentrosymmetric topological
insulators LaBiTe3 and LuBiTe3 and the trivial insulator BiTeI. We find that a robust Weyl-semimetal phase
exists in the solid solutions LaBi1−x Sbx Te3 and LuBi1−x Sbx Te3 for x ≈ 38.5%–41.9% and x ≈ 40.5%–45.1%,
respectively. A low-energy effective model is also constructed to describe the critical behavior in these two
materials. In BiTeI, a Weyl semimetal also appears with applied pressure, but only within a very small pressure
range, which may explain why it has not been experimentally observed.
DOI: 10.1103/PhysRevB.90.155316
PACS number(s): 73.43.Nq, 73.20.At, 78.40.Kc
I. INTRODUCTION
The significance of topology in determining electronic
properties has became widely appreciated with the discovery
of the integer quantum Hall effect and been highlighted further
by the recent interest in topological insulators (TIs) [1–4].
In topological band theory, a topological index, such as the
Chern number or the Z2 index, is well defined only for gapped
systems, and the topological character is signaled by the
presence of novel gapless surface states which cannot exist in
any isolated two-dimensional (2D) system [3,4]. Recently, the
concept of topological phases is further generalized to threedimensional (3D) bulk gapless systems, whose topological
behavior is protected by lattice translational symmetry, known
as the Weyl semimetal (WSM) [5–9].
A Weyl semimetal is characterized by a Fermi energy
that intersects the bulk bands only at one or more pairs of
band-touching points (BTPs) between nondegenerate valence
and conduction bands. This can occur in the presence of
spin-orbit coupling (SOC), typically in a crystal with broken
time-reversal or inversion symmetry but not both, so that the
pairs are of the form (k0 , −k0 ) in the Brillouin zone (BZ).
The effective Hamiltonian around a single BTP k0 can be
written as H (k) = f0 (k) + f(k) · σ , where f0 and f are scalar
and vector functions, respectively, of wave vector in the BZ
and the σj are the Pauli matrices acting in the two-band space.
If one expands the coefficient f(k) to linear order around k0 ,
one gets a Hamiltonian having the form of the Weyl Hamiltonian in relativistic quantum mechanics after a coordinate
transformation in k space. If the sign of the determinant of
the Jacobian that describes the coordinate transformation is
positive (negative), we call the BTP a Weyl node with positive
(negative) chirality, and the low-energy excitations around
such a Weyl node provide a condensed-matter realization of
left-handed (right-handed) Weyl fermions.
These pairs of Weyl nodes are topologically protected in
the sense that they are robust against small perturbations,
which can be seen from the codimension argument as follows.
One can introduce a parameter λ that acts as a perturbation
on the BTP, and let both f0 and f be dependent on λ. In
order to get a band touching at (k0 ,λ0 ), the three coefficients
1098-0121/2014/90(15)/155316(10)
f ≡ (fx ,fy ,fz ) have to vanish. However, since there are four
degrees of freedom, if λ0 → λ0 + δλ, instead of opening a
gap, the Weyl node would just shift slightly in momentum to
compensate for the perturbation. In fact, there is no way to
remove a Weyl node unless two Weyl nodes with opposite
chirality annihilate each other.
If the two Weyl nodes are aligned in energy due to either
time-reversal or some lattice symmetry, and the bands are
filled right up to the Weyl nodes, then the Fermi energy
would be locked there regardless of weak perturbations.
That is, the Fermi level could be slightly shifted upward
(downward) due to some weak perturbation, such that there is
an electronlike (holelike) Fermi surface, then there must also
be a holelike (electronlike) Fermi surface to conserve the total
number of electrons, which is impossible in such a a semimetal.
It follows that the low-energy physics in the Weyl semimetal is
completely dominated by the linearly dispersing states around
the Weyl nodes, which leads to interesting surface states and
transport properties.
The presence of Weyl nodes in the bulk band structure is
responsible for the presence of Fermi arcs at the surface, which
can be understood as follows [7,8]. Consider a small loop in
the 2D surface BZ that encloses the projection along kz of one
Weyl point. When translated along kz , this loop traces out a
surface in the 3D BZ, and the application of Gauss’s theorem
implies that the Chern number on this surface must equal the
chirality of the enclosed Weyl node. It follows that as (kx ,ky )
is carried around the loop, a single electron is pumped up to
(or down from) the top surface, and this is only consistent
with charge conservation if a single surface state crosses the
Fermi energy EF during the cycle. Since this argument applies
for an arbitrary loop, surface states must exist at EF along
some arc emerging from the surface-projected Weyl point. If
there is another Weyl node with opposite chirality, then the
Chern number can vanish once the cylinder encloses both of
the nodes, such that the Fermi arc would only extend between
the two projected Weyl nodes [7,8]. It is also interesting to
note that the surface states residing on the top and bottom
surfaces disperse linearly in opposite directions and remain
robust even when the Fermi energy deviates from the bulk Weyl
nodes [10].
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©2014 American Physical Society
JIANPENG LIU AND DAVID VANDERBILT
PHYSICAL REVIEW B 90, 155316 (2014)
In a WSM with broken time-reversal (TR) symmetry, there
is also a nonzero anomalous Hall conductivity (AHC) that is
closely related to the positions of the Weyl nodes [11–13].
Consider the AHC σxy with the crystal oriented such that the
third primitive reciprocal vector is along z. If we track the
Chern number Cz of a 2D slice normal to z in the 3D BZ, we
must find that it changes by ±1 whenever kz passes a Weyl
node, with the sign depending on the chirality of the node. In
the simplest case, if Cz = 1 for kz between the two Weyl nodes
and zero elsewhere, then the AHC is just proportional to the
separation of the two Weyl nodes in kz . This is also interpreted
as a consequence of the “chiral anomaly” in a WSM [14].
Other interesting transport phenomena can arise due to the
chiral anomaly. For example, if a magnetic field B is applied
to a WSM in the z direction, Landau levels will be formed in the
(x,y) plane. The zeroth Landau level disperses linearly along
kz , but in opposite directions for Weyl nodes with opposite
chirality. As a result, if an electric field E is applied along z,
electrons would be pumped from one Weyl node to the other
at a rate proportional to E · B, with the Fermi arcs serving as
a conduit [5,8,13,15].
As discussed above, a WSM requires the breaking of either
TR or inversion symmetry. Many of the previous works are
focused on WSMs without TR symmetry, such as in pyrochlore iridates [7], magnetically doped TI multilayers [12],
and Hg1−x−y Cdx Mny Te [16]. In this paper, we study the
WSM with preserved TR symmetry but broken inversion
symmetry.
It was argued some time ago that the Z2 -odd and Z2 -even
phases of a noncentrosymmetric insulator should always be
bridged by a critical WSM phase [17,18]. If the transition is
described by some adiabatic parameter λ, then as λ increases
one expects first the appearance of m higher-order BTPs in
the half BZ (and another m at the time-reversed points), where
m = 1 is typical of low-symmetry systems, while m > 1 can
occur when, e.g., rotational symmetries are present. These
higher-order BTPs generally have quadratic dispersion in one
direction while remaining linear in the other two, and are
nonchiral; we refer to such a point henceforth as a “quadratic
BTP.” As λ increases, each quadratic BTP splits to form a pair
of Weyl nodes (4m altogether), which then migrate through
the BZ and eventually annihilate at a second critical value of
λ after exchanging partners. The previous work demonstrated
that this process inverts the strong Z2 index if m is odd [17,18].
Recently, however, Yang et al. claimed that for systems with
certain high-symmetry lines in the BZ, the phase transition
could occur at a unique critical value of λ at which the
bands would touch and immediately reopen, instead of over
some finite interval in λ, even when inversion symmetry is
absent [19]. These authors suggested that BiTeI under pressure
could serve as an example to support their claim [19,20].
In this paper, we address this issue carefully. We show that
an intermediate critical WSM phase should always exist for
any topological phase transition (TPT) between a normal and
a Z2 -odd insulating phase. We find, however, that the width
of the critical WSM phase can be sensitive to the choice of
path in parameter space and can sometimes be very small. To
justify our conclusions, we take specific materials as examples.
We first study the TPT in the solid solutions LaBi1−x Sbx Te3
and LuBi1−x Sbx Te3 using the virtual crystal approximation,
where the phase transition is driven by Sb substitution. The
parent compounds at x = 100%, LaBiTe3 and LuBiTe3 , are
hypothetical noncentrosymmetric materials that are predicted
to be strong topological insulators in Ref. [21] and in the
present work, respectively. Instead, the end members LaSbTe3
and LuSbTe3 at x = 0% are trivial insulators [21]. We find that
a WSM phase is obtained when x is in the range of about
38.5%–41.9% for LaBiTe3 and 40.5%–45.1% for LuBiTe3 .
We further construct a low-energy effective model to describe
the topological and phase-transitional behavior in this class
of materials. We also revisit the TPT of BiTeI driven by
applied pressure, where a WSM phase has not previously
been observed [20,22]. Based on our calculations, we find
that a small interval of WSM phase does actually intervene as
increasing pressure drives the system from the trivial to the
topological phase.
The paper is organized as follows. In Sec. II we derive the
general behavior of TPTs in noncentrosymmetric insulators
and point out some deficiencies in the discussion of BiTeI
by Yang et al. [19] In Sec. III we describe the lattice
structures and basic topological properties of the materials, as
well as the numerical methods used in the realistic-material
calculations, especially the methods used in modeling the
alloyed and pressurized systems and in searching for BTPs in
the BZ. In Sec. IV we present the results for LaBi1−x Sbx Te3 ,
LuBi1−x Sbx Te3 , and BiTeI, and discuss the sensitivity to the
choice of path. In Sec. V, we summarize our work.
II. TOPOLOGICAL TRANSITION IN
NONCENTROSYMMETRIC INSULATORS
A. General behavior
We consider the problem of TPTs in noncentrosymmetric
insulators in the most general case. In the space of the two
bands which touch at the TPT, the system can be described by
the effective Hamiltonian
H (k,λ) = fx (k,λ)σx + fy (k,λ)σy + fz (k,λ)σz ,
(1)
where λ is the parameter that drives the TPT and σx,y,z are
the three Pauli matrices defined in the space spanned by the
highest occupied and the lowest unoccupied states at k. Since
we study the TPT between two insulating phases, we can
assume without loss of generality that the system is gapped
for λ < λ0 , and that the first touching that occurs at λ = λ0
takes place at k = k0 . In other words, fi (k0 ,λ0 ) = 0,i = x,y,z.
Then we ask what happens if k0 → k0 + q and λ0 → λ0 + δλ.
We first expand the coefficients f around (k0 ,λ0 ) as f =
J · q + δλ, where q = k − k0 , δλ = λ − λ0 , J is the Jacobian
with matrix elements Jij = (∂fi /∂kj )|k0 ,λ0 , and is a three
vector with components i = (∂fi /∂λ)|k0 ,λ0 . A natural set
of momentum-space coordinates can be defined
in terms
of the eigensystem J · vi = Ji vi . Defining q = i pi vi and
ui = Ji vi , we obtain
f=
3
pi ui + δλ .
(2)
i=1
Following the argument of Yang et al. [19], the Jacobian
matrix J has to be singular at (k0 ,λ0 ) because otherwise
there would be band touching even when λ < λ0 , contradicting
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PHYSICAL REVIEW B 90, 155316 (2014)
the assumption that the system is insulating for λ < λ0 . This
implies that at least one of the eigenvalues of J is zero. We
assume for the moment that the others are nonzero, i.e., that J
is rank two, and let it be the first eigenvalue that vanishes. Since
the p1 -dependence of f then vanishes at linear order, we again
follow Ref. [19] by including a second-order term to obtain
f = p2 u2 + p3 u3 + δλ + p12 w,
(3)
where w = (1/2)∂ 2 f/∂ 2 p1 |k0 ,λ0 . Now we also have the
freedom to carry out an arbitrary rotation in the pseudospin
representation of the two-band space. That is, we redefine fi
to be the component in the pseudospin direction ei , with e3
given by (u2 ×u3 )/|u2 ×u3 |, and e1 and e2 chosen to form an
orthonormal frame with e3 . Then u23 and u33 vanish, and we
can write explicitly that
f1 = p2 u21 + p3 u31 + δλ 1 + p12 w1 ,
f2 = p2 u22 + p3 u32 + δλ 2 + p12 w2 ,
(4)
f3 = δλ 3 + p12 w3 .
We assume 3 /w3 < 0, since otherwise there are solutions
at negative
√ λ. Then at positive λ, there are always two solutions
p1 = ± −δλ3 /w3 at which f3 = 0. Plugging this into the
expressions for f1 and f2 in Eq. (4), we can obtain p2 and p3
by solving the linear system
u21 u31 p2
1 − w1 3 /w3
+
δλ = 0.
(5)
u22 u32 p3
2 − w2 3 /w3
Solutions of the above equation always exist as long as the
Jacobian matrix J is of rank two, which means a critical WSM
should always exist in the absence of a special lattice symmetry
that would lower the rank of J. From the above it also follows
that at the critical λ = λ0 the dispersion around k0 is quadratic
in p1 and linear in p2 and p3 , and that
√ for larger λ the Weyl
point displacements scale like |p1 | ∼ δλ and |p2,3 | ∼ δλ. The
same conclusions in the rank-two case have been obtained by
Murakami et al. [17] and restated by Yang et al. [19].
If the Jacobian matrix J turns out to be rank one instead
at λ0 , then the bands would first close at a doubly quadratic
BTP. That is, there would be two vanishing eigenvalues of the
Jacobian matrix (which we take to be the first and second), and
the dispersion would be quadratic in p1 and p2 and linear in
p3 . This implies that only the second-order terms associated
with p1 and p2 need to be included in Eq. (3), yielding
f1 = p3 u31 + δλ 1 + p12 w111 + p22 w122 + 2p1 p2 w112 ,
f2 = p3 u32 + δλ 2 + p12 w211 + p22 w222 + 2p1 p2 w212 ,
(6)
f3 = p3 u33 + δλ 3 + p12 w311 + p22 w322 + 2p1 p2 w312 ,
where wij = (1/2)∂ 2 f/∂pi ∂pj |k0 ,λ0 (i,j = 1,2). We can make
a similar transformation on f such that the f3 direction is
e3 = (u3 ×w22 )/|u3 ×w22 |, so that f3 becomes independent of
p3 and p22 . Then one also has the freedom to rotate the p1
and p2 components to make w312 vanish. After these two
transformations, f3 only depends
on p12 and δλ, and one
√
expects solutions at p1 = ± −δλ3 /w311 . Plugging this into
the expressions for f1 and f2 in Eq. (6), one obtains
a quadratic
√
equation for p2 of the form aδλ + bp22 + c δλp2 = 0, where
a, b, and c are some constants determined by the components
of u3 , , and wij (i,j = 1,2). If there are real solutions for the
above equation, then the doubly quadratic BTPs would √
split
into four Weyl
√ nodes whose trajectories scale as p1 ∼ ± δλ
and p2 ∼ ± δλ, p3 ∼ δλ. Otherwise, if there is no solution
for p2 , a gap would be opened up immediately after the band
touching at (k0 ,λ0 ), which would represent the rare case of an
“insulator-insulator transition” using the language of Ref. [19].
However, we do not expect that the strong Z2 index would
be inverted for such an insulator-insulator transition in the
rank-one case. This can be seen as follows. If the BTP does not
lie in any of the TR-invariant slices (kj = {0,π }, j = 1,2,3),
then certainly the 2D Z2 indices of the TR invariant slices
would not change, and it follows that none of the four 3D Z2
indices would change either. If the BTP happens to reside in
one of the TR invariant slices, then since the dispersion in the
2D slice must be quadratic in at least one direction, it should
be topologically equivalent to the superposition of an even
number of linearly dispersing Weyl nodes, which is also not expected to flip the 2D Z2 index, as argued in Ref. [17]. Thus none
of the 3D Z2 indices, including the strong index, would change.
To summarize this section, we find without any latticesymmetry restriction that a critical WSM phase always exists in
the rank-two case. In the rank-one case, an insulator-insulator
type transition is allowed in principle, but would not be
expected to be accompanied by a change in the strong Z2 index.
Therefore, it is fair to claim that, regardless of special lattice
symmetry, there is always a WSM phase connecting Z2 -odd
and Z2 -even phases in a noncentrosymmetric insulator.
B. Discussion of BiTeI
In this section we discuss the TPT in pressured BiTeI, a case
in which the TPT is driven in a system with C3v symmetry.
Contrary to the conclusions of Ref. [19], here we argue that
a critical WSM does exist in the TPT of BiTeI, although the
pressure interval over which it occurs may be rather narrow.
In Refs. [19,20] the authors argued that if there exists
a high-symmetry line in the BZ such that the dispersion
extremum evolves along the line as a function of the adiabatic
parameter (pressure), then one could get an insulator-insulator
type transition without going through a critical WSM. The
authors further pointed out that the high-symmetry lines from
A to H in the BZ of BiTeI, shown in Fig. 1(d), satisfy some
necessary conditions for this to occur. Moreover, they showed
that the symmetry of BiTeI is such that if one concentrates
on the band dispersions along these A–H lines, one finds
a pair of extrema (one valence-band maximum and one
conduction-band minimum) which migrate along the A-H line
as a function of the external parameter (pressure), coincide at
a critical value, and then separate again to reopen the gap.
They furthermore showed that the dispersions are quadratic
in the two orthogonal directions (except exactly at the critical
value), raising the possibility that the extrema in question could
be minima and maxima in all three k-space directions. This
would correspond to the insulator-insulator transition without
an intervening WSM phase. However, our analysis in the
previous section shows that this cannot occur in the rank-two
case, and that the extrema in question actually become saddle
points after the band touchings occur along the A-H lines. In
this case, as recognized in Ref. [19], a WSM phase does occur.
155316-3
JIANPENG LIU AND DAVID VANDERBILT
PHYSICAL REVIEW B 90, 155316 (2014)
In the following section, we will study the TPTs in
various inversion asymmetric materials by first-principles
calculations. We predict LaBi1−x Sbx Te3 and LuBi1−x Sbx Te3
to be WSM candidates within a certain range of impurity
composition x. We also revisit the case of BiTeI, and find that
a WSM phase emerges when external pressure is applied to
BiTeI, but only within a small pressure interval.
III. PRELIMINARIES
A. Lattice structures and basic topological properties
FIG. 1. (Color online) (a) The lattice structure of LaBiTe3 ,
LuBiTe3 , LaSbTe3 , and LuSbTe3 . (b) The BZ of La(Lu)Bi(Sb)Te3 .
(c) The lattice structure of BiTeI. (d) The BZ of BiTeI.
As has been verified in Ref. [19], the Jacobian does remain
of rank two on these lines in BiTeI, and we shall show below
in Sec. IV B that an intermediate WSM phase does occur. We
also point out that Fig. 2 of Ref. [19] does not demonstrate the
absence of the Weyl nodes, since they are expected to lie off
the (kx ,kz ) plane on which the dispersion was plotted.
Yang et al. [19] gave another argument in favor of the
insulator-insulator scenario in BiTeI as follows. They noted
that the band touching first takes place on the A-H line,
which is invariant under the combination of time-reversal and
mirror operations. This imposes some constraints on the form
of the effective Hamiltonian around the BTP, and from these
the authors concluded that, if Weyl nodes do appear, they
should migrate along trajectories of the form p1 ∼ ±δλ1/2 ,
p2 ∼ ±δλ3/2 , and p3 ∼ δλ. Such a curve in 3D space possesses
nonzero torsion, so that the trajectories of the two Weyl nodes
emerging from one quadratic BTP could never join again
and form a closed curve. This implies that if the WSM is
formed by such an event, then it would remain permanently,
contradicting the fact that BiTeI clearly becomes a globally
gapped TI at higher pressures. Based on this reasoning, they
concluded that the TPT in BiTeI must be an insulator-insulator
transition without an intermediate WSM.
However, this argument neglects the fact that the C3v
symmetry means that there are several A-H lines in the BZ of
BiTeI, and the gap first closes by the simultaneous appearance
of quadratic BTPs at equivalent positions on all of these lines.
Even though the two Weyl nodes which emerge from a single
quadratic BTP cannot meet each other, as shown from the
torsion of their trajectories, the Weyl nodes from different
BTPs can interchange partners and eventually annihilate each
other in such a way as to form a closed curve in the BZ. This
is exactly the mechanism of the topological phase transition
in noncentrosymmetric TIs [17,18]. As will be discussed in
Sec. IV B, there are actually six quadratic BTPs in the full
BZ that appear simultaneously, according to the crystalline
and TR symmetries. These six Dirac nodes split into 12 Weyl
nodes, which are eventually gapped out by annihilation after
exchanging partners.
The assumed crystal structures of LaBiTe3 and LuBiTe3
are very similar to Bi2 Te3 , where five atomic monolayers
stack in the [111] direction in an A-B-C-A-... sequence
forming quintuple layers (QLs) as shown in Fig. 1(a). The only
difference is that one of the two Bi atoms in the primitive unit
cell is replaced by a La or Lu atom, which breaks the inversion
symmetry. The lattice structure of LaSbTe3 and LuSbTe3 is
the same as for LaBiTe3 and LuBiTe3 , except that all the Bi
atoms are substituted by Sb. The in-plane hexagonal lattice parameters for LaBiTe3 and LuBiTe3 are a = 4.39 Å and 4.18 Å,
respectively, while the size of a QL along c is 10.07 and 10.29
Å, respectively. The lattice parameters of LaSbTe3 are slightly
different from LaBiTe3 , with a = 4.24 Å and c = 10.13 Å. The
lattice parameters for LuSbTe3 have not been reported before,
so we use those from LuBiTe3 . Among these four hypothetical
materials, LaBiTe3 has been previously reported as a candidate
for an inversion-asymmetric TI [21]. LuBiTe3 is first reported
as a TI candidate in this paper; the nontrivial band topology is
confirmed by calculating the bulk Z2 index [23] and checking
the existence of topological surface states. On the other hand,
LaSbTe3 and LuSbTe3 are trivial insulators.
As shown in Fig. 1(c), BiTeI has a hexagonal lattice
structure with three atoms in the primitive cell stacked as
A-B-C-A-... along the z direction. The lattice parameters in
plane and along the hexagonal axis are a = 4.339 Å and
c = 6.854 Å. BiTeI itself is a trivial insulator with a large
Rashba spin splitting in the bulk [24], but it can be driven
into a TI state by applying pressure. Previous studies have
suggested that the transition to the topological phase is not
mediated by a WSM phase [19,20], but we revisit this issue in
Sec. IV B and come to different conclusions.
B. First-principles methodology
We carry out the bulk first-principles calculations using
the VASP package including SOC [25,26]. The generalizedgradient approximation is used to treat the exchangecorrelation functional [27,28]. The BZ is sampled on an
8×8×8 Monkhorst-Pack [29] k mesh and an energy cutoff of
340 eV is used. The output from the first-principles plane-wave
calculations are then interfaced to the WANNIER90 package [30]
to construct realistic tight-binding (TB) models for these
materials [31].
To describe the electronic structure of LaBi1−x Sbx Te3 and
LuBi1−x Sbx Te3 , we adopt the virtual crystal approximation
(VCA) in which each Bi or Sb is replaced by a “virtual”
atom whose properties are a weighted average of the two
constituents. The VCA treatment typically gives a reasonable
description for solid-solution systems in which the dopant and
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WEYL SEMIMETALS FROM NONCENTROSYMMETRIC . . .
PHYSICAL REVIEW B 90, 155316 (2014)
(a)
10
Gap(meV)
8
6
4
2
0
(b) 8
Gap(meV)
6
4
2
0
0.38
0.40
0.42
Composition x
0.44
0.46
FIG. 2. (Color online) Smallest direct band gap in the BZ vs
composition x for (a) LaBi1−x Sbx Te3 and (b) LuBi1−x Sbx Te3 . Dashed
gray line marks our chosen threshold of 0.2 meV to signal a gap
closure.
To illustrate the topological character, we further calculate
the chirality of the BTPs, which is given by the determinant
of the Jacobian matrix Jij = ∂fi /∂kj . Figure 3 shows how
det(J) varies with x for the BTPs in LaBi1−x Sbx Te3 and
LuBi1−x Sbx Te3 . The red and black open circles mark the
values of det(J) for the BTPs with positive and negative
chirality, which are mapped into each other by mirror
operations about the kx = 0 and other equivalent mirror planes.
One can see that at the beginning of the band touching, the
chirality starts at zero, indicating the creation of a quadratic
(a)
0.5
det (J)
host atoms have a similar chemical character. For example, the
VCA was shown to work well in describing Sb substitution
in Bi2 Se3 , because of the similar orbital character of Sb
5p and Bi 6p, but not for In substitution, where In 5s
orbitals become involved [32]. The VCA is implemented in
the Wannier basis by constructing separate 36-band models
for LaBiTe3 (LuBiTe3 ) and LaSbTe3 (LuSbTe3 ), including
all the valence p orbitals of the cations and anions, as well
as the 5d and 6s orbitals of the rare-earth elements [33]. In
the solid solution, the Hamiltonian matrix elements are then
taken as a linear interpolation in impurity composition x of the
corresponding matrix elements of the parent materials. That
VCA
Bi
Sb
Bi
Sb
is, we take Hmn
= (1 − x)Hmn
+ xHmn
, where Hmn
and Hmn
denote the matrix elements of the TB models of LaBiTe3 and
LaSbTe3 . It worth noting that when generating the Wannier
functions (WFs) for the VCA treatment, the Wannier basis
functions have to be chosen as similar as possible before the
averaging [32]. We therefore use WFs that are constructed
simply by projecting the Bloch states onto the same set
of atomiclike trial orbitals without applying a subsequent
maximal-localization procedure [34,35].
Similarly, to study the pressure-induced TPT in BiTeI,
we carry out first-principles calculations for the system at
the zero-pressure volume, where it is topologically normal,
and also at 85.4% of the original volume, a value chosen
somewhat arbitrarily to be well inside the TI region [20].
We denote these two states as η = 0 and η = 1, respectively.
Then from the Wannier representation we again construct a
realistic Hamiltonian for each system, denoted as H0 and
H1 , respectively, including all the valence p orbitals of Bi,
Te, and I. Finally we linearly interpolate these as H (η) =
(1 − η)H0 + ηH1 , treating η as an adiabatic parameter that
tunes the system through the topological phase transition.
Using these Wannierized effective TB models, we can
search for BTPs very efficiently over the entire BZ. We first
sample the irreducible BZ using a relatively sparse k mesh,
e.g., 20×20×20, and find the point k0 having the smallest
direct band gap on this mesh. A second-round search is
conducted by scanning over a denser k mesh within a sphere
centered on k0 . We then repeat the procedure iteratively until
convergence is reached. All of the trajectories of Weyl nodes
presented in Sec. IV are obtained using this approach.
IV. RESULTS
0
−0.5
A. LaBi1−x Sb x Te3 and LuBi1−x Sb x Te3
(b)
1. Band gap and Weyl chirality
0.5
det (J)
For each of these materials we scan over a mesh in
composition x, and for each x we construct the Wannierized
Hamiltonian for the corresponding solid solution within the
VCA. We then use the methods of the previous section to
search for the BTPs in the entire irreducible BZ. Plots of
the smallest direct band gap in the BZ vs x are presented in
Fig. 2. Clearly the gap remains closed over a finite range of
x in both cases, from 38.5% to 41.9% for LaBi1−x Sbx Te3
and 40.5%to 45.1% for LuBi1−x Sbx Te3 . By checking the
dispersion around the gap-closure point, we confirm that the
system is semimetallic with the Fermi level lying at a set of
degenerate Weyl BTPs over this entire range.
0
−0.5
0.39
0.40
0.41
0.42
0.43
0.44
0.45
Composition x
FIG. 3. (Color online) Determinant of the Jacobian matrix evaluated at Weyl nodes with positive (red) and negative (black) chirality
vs composition x, for (a) LaBi1−x Sbx Te3 and (b) LuBi1−x Sbx Te3 .
155316-5
JIANPENG LIU AND DAVID VANDERBILT
PHYSICAL REVIEW B 90, 155316 (2014)
BTP. As x increases, each quadratic BTP splits into two Weyl
nodes with opposite chirality. These then migrate through the
BZ and eventually annihilate each other at the point where the
chirality returns to zero.
Finally, because of TR symmetry, each Weyl node at k is
always accompanied by another at −k with the same chirality,
giving six more Weyl nodes in the lower half BZ. We thus
generically expect a total of 12 Weyl nodes in the entire BZ
for compositions x in the region of the WSM phase.
2. Symmetry considerations
As mentioned earlier, the point group of this class of
materials is C3v , which has a threefold rotation axis along kz
and three mirror planes that contain the kz axis and
√ intersect the
kz = 0 plane on the lines kx = 0 and ky = ±kx / 3. We define
an azimuthal angle θ that measures the rotation of (kx ,ky ) from
the +ky axis in the clockwise direction as shown in Fig. 4(a).
As a result of the threefold rotational symmetry, if a Weyl
node with positive chirality appears at some θ in the region
0 θ π/3 and at some kz in the upper half BZ, then there
must be another two nodes with the same chirality and the same
kz located at θ + 2π/3 and θ − 2π/3. Taking into account the
mirror symmetry, these must have negative-chirality partners
at the same kz but at −θ , −θ + 2π/3, and −θ − 2π/3.
(a)
ky
0.05
0
3. Weyl trajectories
Figure 4 shows the trajectories of the Weyl nodes in
LaBi1−x Sbx Te3 and LuBi1−x Sbx Te3 projected onto the (kx ,ky )
plane as x passes through the critical region. The red dashed
line represents the trajectory of Weyl nodes with positive
chirality, while the solid black one denotes those with negative
chirality, and the “*” and “⊕” denote the creation and
annihilation points of the Weyl nodes, respectively. As x
increases, six quadratic BTPs are simultaneously created in the
mirror planes; this occurs at xc1 = 38.5% for LaBi1−x Sbx Te3
and 40.5% for LuBi1−x Sbx Te3 . Each quadratic BTP then splits
into two Weyl nodes of opposite chirality, and these 12 nodes
migrate along the solid black and dashed red lines shown in the
figure. Eventually, after exchanging partners, the Weyl nodes
meet and annihilate each other in another set of high-symmetry
planes (ky = 0 and other equivalent planes), at xc2 = 41.9% for
LaBi1−x Sbx Te3 and 45.1% for LuBi1−x Sbx Te3 .
Figures 5(a) and 5(b) show the trajectory of the Weyl nodes
in the kz direction. At x = xc1 , six quadratic BTPs are created,
three in the top half-BZ and three in the bottom half-BZ, but
all of them fairly close to the BZ boundary plane at kz = ±π/c.
As x increases, the six BTPs split to form 12 Weyl nodes, and
these begin to move toward the above-mentioned BZ boundary
plane. Finally, after interchanging partners, Weyl nodes of
opposite chirality annihilate in pairs at xc2 on the BZ boundary
−0.05
(a)
−0.05
0
kx
0.02
0.01
0.01
0
0
−0.01
−0.01
(b)
0.05
(b)
0.02
0.05
−0.02
ky
(c)
−0.02
(d)
0.44
0.44
0
0.42
0.40
0.40
−0.05
−0.05
0
kx
0.05
FIG. 4. (Color online) Trajectories of Weyl nodes in the (kx ,ky )
plane (in units of Å−1 ). Dashed red lines indicate Weyl nodes of
positive chirality; solid black lines are negative. The “*” and “⊕”
denote, respectively, the points of creation or annihilation of Weyl
nodes. (a) For LaBi1−x Sbx Te3 . (b) For LuBi1−x Sbx Te3 .
FIG. 5. (Color online) (a), (b) Trajectories of Weyl nodes in
the kz direction (in units of Å−1 ) for (a) LaBi1−x Sbx Te3 and
(b) LuBi1−x Sbx Te3 . Dashed red (solid black) lines refer to the Weyl
nodes with positive (negative) chirality. θ is the azimuthal angle in
the (kx ,ky ) plane, as indicated in Fig. 4(a). The “*” and “⊕” denote
the creation and annihilation point of the Weyl nodes, respectively.
(c), (d) Trajectories of Weyl nodes in the direction of impurity
composition x for (c) LaBi1−x Sbx Te3 and (d) LuBi1−x Sbx Te3 .
155316-6
WEYL SEMIMETALS FROM NONCENTROSYMMETRIC . . .
PHYSICAL REVIEW B 90, 155316 (2014)
plane at kz = ±π/c. For x > xc2 a global gap opens up and
the system is again an insulator but with an inverted Z2 index.
The locus of Weyl points can be regarded as forming a loop
in the four-dimensional space of (kx ,ky ,kz ,x), and just as this
loop can be projected onto kz as in Figs. 5(a) and 5(b), it can
also be projected onto the direction of impurity composition
x as shown in Figs. 5(c) and 5(d). Again, it is clear that
the Weyl nodes are created at xc1 in the mirror planes and
annihilated at xc2 at θ = ±π/6. These plots may also be
helpful in seeing how the high sixfold symmetry contributes
to the narrowness of the WSM region. If the symmetry of the
system were lower, the period of oscillation in θ in Figs. 5(c)
and 5(d) would be longer, which would allow the Weyl nodes
to oscillate farther in the x direction, giving a wider window
of concentration for the WSM phase. In contrast, a fictitious
system with an N -fold rotational symmetry would force the
width of the WSM region to vanish as N → ∞. Here we have
N = 6, which is evidently large enough to limit the WSM
phase to a rather small interval in x.
4. Surface Fermi arcs
One of the most characteristic features of WSMs is the
existence of Fermi arcs in the surface band structure. Here we
calculate the surface states using the surface Green’s-function
technique [36], which is implemented in the context of the
VCA effective Hamiltonian in the Wannier basis. The surface
BZ is sampled by a 64×64 k mesh, and the surface spectral
functions calculated on this mesh are then linearly interpolated
to fit a 128×128 k mesh. Figure 6 shows the normalized
surface spectral functions averaged around the Fermi level
for LaBi1−x Sbx Te3 at x = 0.405 and for LuBi1−x Sbx Te3 at
x = 0.43. The averaging is done over an energy window of
±4.5 meV around the Fermi energy, which is determined
by the position of the bulk Weyl nodes. Six Fermi arcs
connecting the projected Weyl nodes of opposite chirality are
visible, confirming the existence of the WSM phase in these
two solid-solution systems. Note that because of the small
projected bulk gap on the loops where the Fermi arcs reside,
some non-negligible spectral weight is visible even outside
the Fermi arcs in Fig. 6, coming from the artificial smearing
of the Green’s functions.
5. Simplified six-band model
In order to capture the essential physics in these materials,
we construct a six-band TB model to describe the interesting
critical behavior. From the band-structure plots presented in
Fig. 7, it is clear that the band inversion occurs around the
Z point of LaBiTe3 and LuBiTe3 , so we focus our attention
on the six states at Z closest to the Fermi level. A symmetry
analysis shows that these six states belong to two copies of
the two-dimensional Z6 irreducible representation (irrep) of
the C3v group at Z, plus a Kramers pair of one-dimensional
complex-conjugate Z4 and Z5 irreps corresponding to linear
combinations of jz = ±3/2 orbitals.
We thus build our six-band TB model out of basis states
having the symmetry of |pz , ↑
and |pz , ↓
on the Te
atoms at the top and bottom of the quintuple layer, and
(a) 2
(a)
1
0.05
1
Energy(eV)
0.8
ky
0.6
0
0.4
0
−1
0.2
−0.05
−2
(b)
1
(b) 2
0.05
0.8
1
Energy(eV)
ky
0.6
0
0.4
0.2
−0.05
0
−1
−0.05
0
kx
0.05
−2
FIG. 6. (Color online) Surface spectral function averaged around
the Fermi level (kx and ky in units of Å−1 ) for (a) LaBi1−x Sbx Te3 at
x = 0.405 and (b) LuBi1−x Sbx Te3 at x = 0.43.
FIG. 7. (Color online) Bulk band structures of (a) LaBiTe3 and
(b) LuBiTe3 .
155316-7
JIANPENG LIU AND DAVID VANDERBILT
PHYSICAL REVIEW B 90, 155316 (2014)
TABLE I. Parameters of the six-band model (in eV).
0.2 − δ/4
0.15 − δ/4
δ
0.1 − δ/4
0.24 − δ/2
0.2 − δ/2
t1
t2
t3
t4
λ1
λ2
FIG. 8. (Color online) Top: Schematic diagram of the interlayer
spin-independent hopping terms in the six-band model. Orbitals on
sites Te1, Te2, and Te1 make up a quintuple layer; A, B, and C
label in-plane hexagonal positions. Bottom: Phase diagram for the
topological behavior of the six-band model.
0.15 − δ/2
0.12 − δ/2
0.12 + δ/2
0.06 − δ/2
0.05
0.1
λ3
λ4
tu
tv
v1
v2
v3
E1
E2
E3
0
0.1 + δ − 6v1
−6v2
−0.1 − δ
Figs. 9(c) and 9(d). These Weyl nodes eventually annihilate
with each other at δ ≈ 0.074 eV after exchanging partners,
which qualitatively reproduces the phase-transition behavior
of the VCA effective Hamiltonians very well. When δ > 0.074
eV, the system becomes a strong TI. The bulk band structure
at δ = 0.09 eV in the TI phase is shown in Fig. 9(a), which
very well captures the low-energy dispersions around Z that
were found in the first-principles calculations.
6. Discussion
To conclude this section, we would like to comment that
the width of the WSM phase depends on two ingredients. On
one hand, as discussed above, it depends on the symmetry of
the system; other things being equal, the WSM interval tends to
be wider in systems with lower symmetry. On the other hand,
even for fixed symmetry, it also depends on the the detailed
choice of path connecting the topological and trivial phases.
Choosing a different path may broaden or reduce the WSM
region. For example, if one artificially changes the strength
of the atomic SOC strength in LaBiTe3 and LuBiTe3 in the
Wannierized TB models, and scales the variation of the actual
atomic SOC by a single scaling parameter λ, then we find
(a)
(b)
6
Gap (meV)
1
Energy (eV)
|px + ipy , ↑
and |px − ipy , ↓
combinations located on the
central Te atoms. A schematic illustration of the six-band
model is shown in Fig. 8, where the top, bottom, and central
Te atoms are denoted by Te1, Te1 , and Te2, respectively.
First of all, six interlayer spin-independent hopping terms are
included in the model. As shown in Fig. 8, we consider the
first-neighbor hopping between the central and top (bottom)
Te atoms t1 (t2 ), the inter-QL (intra-QL) hopping between the
top and bottom Te atoms t3 (t4 ), and some further-neighbor
hoppings tu and tv that are crucial in obtaining a nontrivial
Z2 index. Second, to capture the Rashba spin- splitting in
the first-principles band structure, in-plane Rashba-like spindependent hoppings within the top and bottom Te monolayers
are included and are denoted by λ1 and λ2 , respectively. For
completeness, the interlayer first-neighbor Te1-Te2 (λ3 ) and
Te1 -Te2 (λ4 ) Rashba-like hopping terms are also included.
Lastly, to reproduce the first-principles band structure better,
we also introduce first-neighbor spin-independent hopping
terms within the Te1, Te2, and Te1 monolayers, denoted by v1 ,
v2 , and v3 , respectively. The on-site energies are also different
and are labeled by E1 for Te1, E2 for Te2, and E3 for Te1 . As
our model is only intended to be semiquantitative, we use the
same model parameters to describe both LaBiTe3 and LuBiTe3 .
We take all of the parameters in the model to depend on a
scaling parameter δ that drives the TPT. When δ is zero, the
system is a trivial insulator; as δ increases, the system becomes
a topological insulator by going through a critical WSM. The
dependence of the parameters on δ defines a path in parameter
space. It is important to note that the width of the critical WSM
region can be highly sensitive to this path, with an improper
choice sometimes leading to an extremely narrow WSM phase.
Our choice is specified in Table I.
Following the path we have chosen, a WSM phase is
obtained for 0.067 eV < δ < 0.074 eV. As shown in Fig. 9(b),
the smallest direct band gap in the BZ vanishes when 0.067 eV
< δ < 0.074 eV, indicating the existence of BTPs in BZ. If one
further checks the position of the BTPs, one finds that when
δ ≈ 0.067 eV, six quadratic BTPs are created in the mirror
planes, which then split into 12 Weyl nodes and propagate
in the BZ following the solid black and dashed red lines in
0
4
2
−1
0
(c)
0.067
0.070
0.073
(d)
0.1
0.05
0.05
0
0
−0.05
−0.1
−0.1 −0.05
−0.05
0
0.05
0.1
FIG. 9. (Color online) (a) Bulk bandstructure of the six-band
model at δ = 0.09 eV. (b) Smallest direct band gap in the BZ vs. δ.
(c) Trajectory of Weyl nodes projected onto the (kx ,ky ) plane. Dashed
red (solid black) line refers to the Weyl node with positive (negative)
chirality. The “*” and “⊕” denote the creation and annihilation point
of the Weyl nodes respectively. θ is the azimuthal angle in the (kx ,ky )
plane. (d) Trajectory of Weyl nodes along kz . Units of kx , ky and kz
are Å−1 .
155316-8
WEYL SEMIMETALS FROM NONCENTROSYMMETRIC . . .
PHYSICAL REVIEW B 90, 155316 (2014)
that the WSM region only shows up for λ in the range of
76.8%–77.3%, which is significantly narrower than for the
VCA case. However, if an average SOC is applied to the entire
system, such that the SOC strength on Te is artificially high
and that on Bi is artificially low, we find that a much wider
WSM region results. Thus, it may potentially be possible to
engineer the width of a critical WSM phase if one can modify
the transformation path, as by epitaxial strain, pressure, or
additional chemical?brk?> substitution.
(a)
0.04
(b)
0.015
0
0
−0.04
−0.015
−0.04
B. BiTeI: Revisited
In order to justify the discussion in Sec. II B, we revisit
the TPT in BiTeI driven by pressure. In our calculations, the
pressure is applied by compressing the volume of the primitive
cell. The fully compressed volume V is taken to be 85.4% of
the original volume V0 , such that the former is well inside
the topological region [20], and both the lattice vectors and
atomic positions are relaxed at the compressed volume. As
discussed in Sec. III B, we searched for BTPs over the entire
irreducible BZ for a transitional Hamiltonian scaled as H (η) =
(1 − η)H0 + ηH1 for 0 η 1, where H0 and H1 represent
the Hamiltonians of the uncompressed and fully compressed
BiTeI, with even and odd Z2 indices, respectively. As shown
in Fig. 10(a), as the pressure is increased from 0% to 100%
(alternatively, as V is decreased from 100% to 85.4% of V0 ),
a semimetallic phase emerges for η in the range of about
54%–56%.
(a) 8
7
Gap (meV)
6
5
4
3
2
1
0
0.53
0.54
0.55
0.56
0.57
1
(b)
0.05
0.8
0.6
0
0.4
0.2
−0.05
−0.05
0
0.05
0
FIG. 10. (Color online) (a) Smallest direct band gap in the BZ of
BiTeI vs the pressure-scaling variable η. (b) Surface spectral function
of BiTeI in the WSM phase (at 55% of the full pressure).
0
0.04
FIG. 11. (Color online) (a) Trajectories of Weyl nodes in the
(kx ,ky ) plane (in units of Å−1 ). Dashed red (solid black) lines indicate
the trajectories of Weyl nodes with positive (negative) chirality. The
“*” and “⊕” denote the creation and annihilation point of the Weyl
nodes, respectively. (b) Trajectory of Weyl nodes in the kz direction
(units of Å−1 ).
The point group of BiTeI is the same as for LaBiTe3 and
LuBiTe3 , namely, C3v . Therefore, as explained in Sec. IV A,
one would expect the emergence of 12 Weyl nodes in the
entire BZ during the phase-transition process. The trajectories
of the Weyl nodes are plotted in Figs. 11(a) and 11(b).
When η ≈ 54%, six quadratic BTPs are first created at the
BZ boundary kz = π/c in the ky = 0 and other equivalent
high-symmetry planes. These BTPs then split into 12 Weyl
nodes which propagate along the directions indicated by solid
black (antimonopoles) and dashed red (monopoles) lines.
They annihilate each other in the three mirror planes after
exchanging partners. Note that in this case the system goes
from a normal to topological insulator as η increases, which is
the reverse of the LaBi1−x Sbx Te3 and LuBi1−x Sbx Te3 cases.
The results shown in Fig. 11 support our conclusions in
Sec. II B. In particular, even though the torsion argument
implies that the trajectories of the two Weyl nodes which split
off from a given quadratic BTP would never meet each other,
a closed curve is still formed in the 3D BZ of BiTeI through
the interchange of partners among the Weyl nodes.
Figure 10(b) shows the surface spectral function of BiTeI
averaged around the Fermi level for η = 0.55, in the WSM
phase. It is clear that there are six Fermi arcs extending between
the six pairs of projected Weyl nodes, which is again the
hallmark of a WSM phase.
We therefore conclude that a WSM phase does exist in the
TPT of BiTeI, but it occurs only within a narrow pressure range.
If η is changed by 2.5%, the volume is only changed by 0.39%,
which might be difficult to measure experimentally. Again, the
narrowness of the WSM interval can be attributed in part to
the high symmetry of the system. However, as emphasized in
the previous section, the width of the critical WSM is also
sensitive to the choice of path in parameter space. The critical
WSM could get broadened by choosing a different path, as
for example by applying uniaxial pressure. We leave this for a
future study.
V. SUMMARY
In this paper, we have investigated the nature of the TPT
in a noncentrosymmetric TI in the most general case. We find
155316-9
JIANPENG LIU AND DAVID VANDERBILT
PHYSICAL REVIEW B 90, 155316 (2014)
that an intermediate WSM phase is always present, regardless
of other lattice symmetries, as long as inversion symmetry
is absent. We discussed separately the cases in which the
Jacobian matrix is rank one or rank two when the gap first
closes. In the rank-two case, each quadratic BTP would always
split into a pair of Weyl nodes, which annihilate each other after
exchanging partners. If the rank of the Jacobian is one, then
the doubly quadratic BTP in this case would either split into
four Weyl nodes, or else immediately be gapped out again,
corresponding to an “insulator-insulator transition.” However,
in the latter case, the bulk Z2 indices are not expected to
change. Therefore, we conclude that Z2 -even and Z2 -odd
phases of a noncentrosymmetric insulator must always be
separated by a region of WSM phase, even if other symmetries
are present.
To illustrate our conclusions, we have carried out calculations on specific noncentrosymmetric insulators. For
LaBi1−x Sbx Te3 and LuBi1−x Sbx Te3 we have used Wannierized VCA Hamiltonians to find a WSM phase in the region
x ≈ 38.5%–41.9% and x ≈ 40.5%–45.1%, respectively. A
six-band TB model was also constructed to describe the
topological and critical behavior in these materials. We found
that the width of the critical WSM phase can be highly sensitive
to the choice of path in the parameter space, suggesting that
there is flexibility to engineer the WSM phase.
We have also revisited the TPT of BiTeI as a function of
pressure, where previous work suggested the absence of a
WSM phase [19]. Using a carefully constructed algorithm to
search for the minimum gap in the full three-dimensional BZ,
we found that a WSM phase is indeed present over a narrow
interval of pressure, although this range may be so narrow as
to make its experimental observation difficult.
In summary, we have clarified the theory of a general
Z2 -even to Z2 -odd topological phase transition in a threedimensional time-reversal-invariant insulator with broken
inversion symmetry, and demonstrated that an intermediate
WSM phase must always be present. We have also detailed the
behavior of LaBi1−x Sbx Te3 and LuBi1−x Sbx Te3 as promising
candidates for WSMs of this kind. While we have not
considered disorder or interactions explicitly, we expect our
conclusions to survive at least for weak disorder or interactions. Our work is a step forward in the general understanding
of topological phase transitions, and may provide useful
guidelines for the experimental realization of new classes of
Weyl semimentals.
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ACKNOWLEDGMENT
This work is supported by NSF Grants No. DMR-10-05838
and No. 14-08838.
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